From ef963a96ab176ea0c06b3b3ac49306a56d4b3374 Mon Sep 17 00:00:00 2001 From: "lisa.pizzo" Date: Wed, 29 Oct 2025 18:20:30 +0100 Subject: [PATCH] Upload files to "Sheet2" --- Sheet2/ExC_Num.m | 44 ++++++++++++++++++++++++++++++++++++++++ Sheet2/PlotC.jpg | Bin 0 -> 132506 bytes Sheet2/Sheet2_Pizzo.pdf | Bin 0 -> 413975 bytes 3 files changed, 44 insertions(+) create mode 100644 Sheet2/ExC_Num.m create mode 100644 Sheet2/PlotC.jpg create mode 100644 Sheet2/Sheet2_Pizzo.pdf diff --git a/Sheet2/ExC_Num.m b/Sheet2/ExC_Num.m new file mode 100644 index 0000000..bcb91fe --- /dev/null +++ b/Sheet2/ExC_Num.m @@ -0,0 +1,44 @@ +% Parameters +N = 501; % number of grid points +x = linspace(0,1,N); % grid +dx = x(2)-x(1); % grid spacing +p_values = [-10, -3, -1, 0, 1, 3, 10]; + +% Analytical solution function +u_analytical = @(x,p) (abs(p) < 1e-12).*x + (abs(p) >= 1e-12).*((exp(p.*x)-1)./(exp(p)-1)); + +figure; +hold on; + +for k = 1:length(p_values) + p = p_values(k); + + e = ones(N-2,1); + % Second derivative: -u'' + D2 = spdiags([e -2*e e], -1:1, N-2, N-2)/dx^2; + % First derivative: p*u' + D1 = spdiags([-e*0.5 e*0.5], [-1 1], N-2, N-2)/dx; + L = -D2 + p*D1; + + % Right-hand side + b = zeros(N-2,1); + + % Incorporate boundary conditions + b(1) = b(1) + 0/dx^2; % u(0)=0 + b(end) = b(end) + 1/dx^2; % u(1)=1 + + % Solve linear system + u_inner = L\b; + u_num = [0; u_inner; 1]; % add boundary values + + % Plot numerical and analytical solution + plot(x, u_num, '--', 'LineWidth', 2, 'DisplayName', sprintf('num, p=%g',p)); + plot(x, u_analytical(x,p), '-', 'LineWidth', 2, 'DisplayName', sprintf('ana, p=%g',p)); +end + +xlabel('x'); +ylabel('u(x)'); +title('Numerical vs Analytical solution of -u'''' + p u'' = 0'); +legend show; +grid on; +hold off; diff --git a/Sheet2/PlotC.jpg b/Sheet2/PlotC.jpg new file mode 100644 index 0000000000000000000000000000000000000000..dbb9acb77c8543e8468ae38fbda67a833e232e92 GIT binary patch literal 132506 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